Torus Quantization for Spinning Particles

We derive semiclassical quantization conditions for particles with spin. These generalize the Einstein-Brillouin-Keller quantization in such a way that, in addition to the Maslov correction, there appears another term which is a remnant of a non-Abelian geometric or Berry phase. This correction is interpreted in terms of a rotation angle for a classical spin vector.

Figure 1

The invariant manifolds ${\mathcal{T}}_{\theta }$ of $Y_{\mathrm{c}\mathrm{l}}^t$ [see (15)] are given by tori ${\mathbb{T}}^d$ to which at each point is attached the set of all points on the two-sphere $S^2$ sharing a fixed latitude $\theta$ with respect to a varying axis $\mathit{n}(\mathit{p},\mathit{x})$.